Flower Pollination Heuristics for Nonlinear Active Noise Control Systems

: Abstract In this paper, a novel design of the flower pollination algorithm is presented for model identification problems in nonlinear active noise control systems. The recently introduced flower pollination based heuristics is implemented to minimize the mean squared error based merit/cost function representing the scenarios of active noise control system with linear/nonlinear and primary/secondary paths based on the sinusoidal signal, random and complex random signals as noise interferences. The flower pollination heuristics based active noise controllers are formulated through exploitation of nonlinear filtering with Volterra series. The comparative study on statistical observations in terms of accuracy, convergence and complexity measures demonstrates that the proposed meta-heuristic of flower pollination algorithm is reliable, accurate, stable as well as robust for active noise control system. The accuracy of the proposed nature inspired computing of flower pollination is in good agreement with the state of the art counterpart solvers based on variants of genetic algorithms, particle swarm optimization, backtracking search optimization algorithm, fireworks optimization algorithm along with their memetic combination with local search methodologies. Moreover, the central tendency and variation based statistical indices further validate the consistency and reliability of the proposed scheme mimic the mathematical model for the process of flower pollination systems.

The ANC is a fundamental problem in control engineering and has been studied extensively with both traditional and different local/global optimization techniques [36]. The well-known local search methods used in ANC systems are based on the least mean squares approach [37][38][39][40][41][42][43][44]. The local search algorithms are easy to implement but suffer from premature convergence, i.e., local minima issues. To overcome these issues, different global search based soft computing techniques are proposed such as, genetic algorithms (GAs) [45,46], particle swarm optimization (PSO) [47][48][49], backtracking search heuristics [50], fireworks algorithm [51], and artificial neural networks [52]. The optimization strength of FPA looks promising to be exploited for ANC problems as an alternate, accurate, reliable, and robust computing paradigm. The innovative contributions of the current study are given as: • The design of FPA based intelligent computing paradigm is presented for an effective solution of nonlinear ANC systems. • Mean squared error based merit function with nonlinear Volterra series filtering is formulated. • The accurate and robust performance of the FPA based ANC for various noise interferences in the case of different primary and secondary path scenarios prove the efficacy of the approach. • Central tendency and variation based statistical indices validate the consistency and reliability of the proposed scheme.
The rest of the manuscript is prepared as: ANC model is given in Section 2. The design approach is described in Section 3. Section 4 presents the results and the comparative studies with state of the art counterparts, and conclusions are given in Section 5.

System Model: ANC
The conventional block diagram of ANC based controller is given in Fig. 1, and the proposed model of nonlinear ANC with FPA is shown in Fig. 2. The algorithm used for filter's coefficients update belongs to a class of nature-inspired heuristics named FPA. In the proposed mechanism, the reference microphone detects the source noise and error microphone measures the output response of a system. When anti-noise and source noise signals combine silence zone is created. The proposed mechanism works on the principle of superposition theorem [53]. Related information for ANC system model can be seen in [54][55][56] and citations mentioned therein.

Control System
Cancellation Area Figure 2: Process block structures of proposed flower pollination algorithm based ANC system

System Methodology
The methodology for ANC modeling with FPA consists of two phases; (1) formulating fitness function (2) presenting optimization mechanism based on FPA. The detailed flowchart in terms of process block structure is shown in Fig. 3.

Modeling for ANC
Block diagram of proposed ANC controller is given in Fig. 2 while the adjustable parameter L-tap weights, i.e., decision variables of optimization algorithm, for ANC system based on nonlinear filtering with Volterra series is given mathematically as: here b(k) represents coefficients Volterra filter at instance k. Let B be a set of contestant solutions of ANC systems, i.e., elements of FPA, a set of k numbers of b as in (1) construct B as follows The input noise interference or source signal s(k) and output of nonlinear adaptive Volterra filtering b(k) with length L = 20, i.e., VF-T1, for the population B, then ANC system using (1) and (2) is written as: . . .
Accordingly, Volterra filtering of type 2 (VF-T2) with L = 35 for the ANC system using Eqs. (1) and (2) is written as: Similarity, for Volterra filtering of type 3 (VF-T3) in case of the length of the Volterra filter L = 65 in ANC system is given by: In case of [c 1 ,c 2 ,...,c L ] T are the response of secondary path transfer function C(z) with L-tap weights/coefficients is written as: ⎡

The fitness and time of FPA base ANC for all three Volterra filters
Controller re fe re nc e m icr op ho ne error microphone loudspeaker

Histogram analysis for in-depth performance
Initialize populations best solution g with minimum fitness and set the switching probability p  The fitness or merit function for ANC model is given as: for Here v(k) and q * (k) are the desired and estimated responses of the primary and secondary paths, respectively. and is the response of the secondary path. Eq. (7) equivalently represented as: ⎡ In the case of perfect model, one has fitness function u = 0, so optimization mechanism is exploited for tuning of fitness (7), such that the magnitude of residual error of the ANC system is reduced substantially. In the next section, optimization of ANC system with FPA is presented.

Optimization: Flower Pollination Algorithm
The FPA is a mathematical model inspired by the process of pollination dynamics in flowers during the reproduction mechanism [14]. Yang et al. [15] introduced FPA in early 2012 as an alternate optimization solver for both global and local search. Most of the flower plants reproduction strategy is based on the pollination process in which pollen is transferred from one plant to another plant of flowers by butterflies, insects, birds, and bees. The pollination process is segmented into biotic and abiotic types. Biotic type flower pollination is also called crosspollination, i.e., the main form of flowering pollination, in which pollens are transferred by insects and birds. The majority of flowering plants use biotic pollination procedures for pollen spread over a long distance via Lèvy flights. While in abiotic pollination, the flowering plants does not required pollinators and 10% of total flowering plants follow such pollination. In abiotic, the distance covered by the pollinators is short and such types of actions are considered as local search. Biotic and abiotic characteristics of pollinators are used to design an optimization algorithm called FPA. The four basics rules of FPA based heuristic are introduced by Yang in 2012 as follows: Rule 1. Global pollination carried out via biotic/cross pollination procedures with the help of insects, birds and bees to transport the pollens. Rule 2. Abiotic or self-pollination process is adapted for efficient local search. Rule 3. Flower fidelity process based reproduction probability. Rule 4. Switching probability between 0 and 1 is exploited for feasible local and global pollination process [21].
The impressive swarm based optimization characteristics of FPA is exploited by the scholars from different fields [57][58][59][60]. The mathematical mechanism of FPA bases of these four rules are given as follows [25]: where, x t i stands for pollen vector for ith candidate solution at iteration t, g * represents best solution at current iteration and L stands for Lèvy flight represented as: here Γ (λ) represents the gamma function while distribution is effective for s > 0 and λ = 1.5. The local search with FPA is represented as: here, μ be the uniform distribution between 0 and 1, x t j and x t k are jth and kth pollens vectors from different flowers of the same plant, respectively. In this study, the meta-heuristics of FPA based ANC is implemented as shown in Fig. 3, while the pseudocode is presented in Algorithm 1.

Results and Discussion
The results of detailed ANC experimentations are presented here for multiple independent executions of the FPA. Three ANC problems are implemented based on different lengths (L) of Volterra filter (VF), i.e., L = 20,35, and 65 in the case of VF-T1, VF-T2, and VF-T3, respectively. The FPA based ANC system are evaluated for sinusoidal/random/complex random noise interferences having linear primary path (LPP), nonlinear primary path (NPP), linear secondary path (LSP) and nonlinear secondary path (NSP). The transfer function for LPP is: While, in case of LSP, the transfer function is defined as The NPP transfer function is given as: Let q * (k), i.e., an anti-noise signal is generated by the NSP as: The simulations are conducted in Matlab R2017b running under Windows 10 environment on DESKTOP-73HVB7M, with Intel(R) Core(TM) i7-4790 CPU@3.60 GHz, 16-GB RAM.

Problem 1: ANC Model for Sinusoidal, Random and Complex Random Signals of VF-T1
In this problem, FPA based ANC system is exploited for Case 1: ANC for LPP and NSP The performance of the FPA is further examined through histogram plots and statistical measures of minimum (MIN), mean, and standard deviation (STD). The histogram plots are provided in Fig. 6 for all cases of Problem 1. While the statistical operators are given in Tab. 1 and one may observe that the results of random VF-T1 are relatively better than that of sinusoidal, but a bit degraded to complex random. Moreover, the small STD values further validate the precision of the proposed FPS based ANC controllers.
The computational complexity of the FPA based ANC controllers is evaluated via mean time of execution required for the optimization and results for mean along with STD are tabulated in Tab. 2. It is observed that the average time lies around 100±50, 85±80 and 80±50 for sinusoidal VF-T1, random VF-T1, and complex random VF-T1 cases.

Problem 2: ANC Model for Sinusoidal, Random and Complex Random Signals of VF-T2
In problem 2, FPA based ANC system is implemented for Case 1: ANC for LPP and NSP (ANC-LPP-NSP), Case 2: ANC for NPP and LSP (ANC-NPP-LSP) and, Case 3: ANC for NPP and NSP (ANC-NPP-NSP).
Graphical representation of the statistical outcomes for hundred independent trials of the FPA based ANC for each case of different noise interferences are given in Fig. 7. The statistical operators are given in Tab. 3. The fitness values of FPA based ANC system for ANC-LPP-NSP, ANC-NPP-LSP and ANC-NPP-NSP are around 10-05 to 10-06 for sinusoidal, 10-04 to 10-05 for random and 10-04 to 10-06, for complex random noise interferences of VF-T2. The results for different scenarios presented in Fig. 8 show that the proposed FPA controllers are effective for the reliable treatment of ANC systems.
The performance of the FPA based ANC systems is further investigated through histogram plots and STATISTICAL operators and it is observed that the results of random VF-T2 are better than that of complex random but inferior to sinusoidal VF-T2. One may decipher that relatively better accuracy is attained for ANC system based sinusoidal and random noise signals. While the results of ANC with sinusoidal noise are consistently found better than random noise scenarios.

Problem 3: ANC Model for Sinusoidal, Random and Complex Random Signals of VF-T3
In this problem, FPA based ANC system is exploited for different primary/secondary path scenarios. The proposed FPA based ANC are conducted for hundred independent trials and graphical representation of the statistics in sort and unsorted plots are given in Fig. 9. The histograms are provided in Fig. 10 for each case of problem 3, while the statistics are provided in Tab. 5. It is observed that the results of random VF-T3 are better than that of complex random but inferior to sinusoidal VF-T3. Relatively better outcomes in term of accuracy are observed for FPA based ANC system in case of sinusoidal and random noise interferences. The comparison shows that sinusoidal noise interference based ANC with VF-T3 are consistently superior than each random noise scenario.  The computational complexity analyses for the optimization of FPA based ANC is also evaluated based on mean execution time and STD, and results are provided in Tab. 6. The analysis show that the average time lies around 15±10 for sinusoidal, 30±15 for random and 60±40 for complex random noise interferences of ANC with VF-T3 cases, respectively.
The computational complexity of FPA based ANC is examined with counterpart optimization solvers. The computational complexity on mean execution time index of BSA and BSA-SQP results for sinusoidal noise signal are lie around 800±50, 1000±60 and 780±30 for FIR, VF-1 and VF-2, respectively, while 750±100 and 900±90 for random and complex random noise signals, respectively [50]. The complexity of variants of GAs and its moments combination of IPA, i.e., GA-IPM-1 to 12, for ANC with FIR filter with sinusoidal, random and complex random noise interference is 40±10, 60±5, and 90±5, respectively. Computational complexity of nature inspired heuristics of PSO and its hybridized methodologies with PSO-IP, PSO-AS, PSO-SQP, and PSO-NM for ANC system based on FIR filtering for all three noise variations is around 7±4 [49]. The computational complexity on mean values of respective FWA, enhanced FWA and adaptive FWA are around 250±40, 145±40, and 100±20 for sinusoidal noise signal, 530±20, 340±20, and 220±10 for random noise signal and 835±17, 540±25, and 352±5 for complex random signal [51]. One can quite evidently observe that the complexity requirements of FPA based ANC system is relatively superior from GAs, BSA and FWA along with their memetic combination with local search methodologies. While the results of PSO based variants are efficient from rest but these results are for ANC systems based on FIR filtering having relatively inferior in accuracy from FPA based ANC.

Comparative Study with Reported Results
Comparative studies of FPA results for ANC systems are made with reported studies based on adaptive genetic algorithm AGA [58], variants of memetic combination of GAs with interiorpoint (IP) algorithm, i.e., GA-IPA-1, to GA-IPA-12 [59], nature-inspired heuristic via particle swarm optimization (PSO) and its hybrid with IP (PSO-IP), active-set (PSO-AS), sequential quadratic programming (PSO-SQP) and Nelder-Mead (PSO-NM) methods [49], backtracking search optimization algorithm (BSA) and its hybrid with SQP (BSA-SQP) [50], and variants of fireworks algorithm (FWA) [51]. One may decipher from all these reported results and statistical observation that the accuracy and convergence of FPA are in good agreement with state of the art methodologies for all three ANC problems. An additional advantage of FPA based ANC is that optimization of decision variables is based on a standalone algorithm with the ability of both local and global search whereas most of the reported results are based on hybrid methodologies.   Figure 11: Pseudocode for flower pollination algorithm for nonlinear ANC

Conclusions
A novel design of nature-inspired heuristic of FPA is presented for the identification problem in nonlinear ANC with interferences. Different ANC scenarios by considering linear/nonlinear and primary/secondary paths are evaluated by determining coefficients of three different Volterra filters, i.e., VF-T1, VF-T2 and VF-T3. The performance of the FPA based ANC is verified through consistently achieving reasonable gauges of statistical operators in terms of accuracy, convergence and complexity measures. The performance is further validated via histogram analysis to prove that the FPA based ANC systems are reliable, accurate, stable and robust but the performance of the VF-T3 is comparatively better. The accuracy of FPA based ANC is in good agreement with state of the art counterpart solvers based on GA, PSO, BSA and FWA along with their hybrid with local search. In the future, one may explore to enhance the performance of ANC system by implementation of recently introduced fractional derivative definition [60][61][62][63][64]. Moreover, the proposed methodology can be exploited to efficiently solve various complex engineering optimization problems [65][66][67][68].

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.